Fringe Measurement and Control for the Keck Interferometer

Transcription

1 PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC, 122: , 2010 July The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A. Fringe Measurement and Control for the Keck Interferometer M. M. COLAVITA, 1 A. J. BOOTH, 1,2 J. I. GARCIA-GATHRIGHT, 1,3 G. VASISHT, 1 R. L. JOHNSON, JR., 4 AND K. R. SUMMERS 5 Received 2010 April 17; accepted 2010 May 25; published 2010 June 22 ABSTRACT. The paper describes the measurement and control algorithms of the near-infrared fringe trackers of the Keck Interferometer (KI). The fringe trackers provide fringe amplitude measurements for visibility science, and feedback and feedforward signals for fringe tracking and cophasing, respectively. Fringe measurement uses temporal path length modulation with synchronous array readout, and implements bias corrections of both the quadratures and the fringe power to provide high accuracy. The KI implementation uses sliding-window estimation to reduce control latency, and implements signal-to-noise (S/N) estimators that minimize biases from intensity scintillation. An inner-loop/outer-loop controller is used for ordinary fringe tracking, and a discrete-unwrapping controller is used for cophasing. A robust algorithm is used to estimate the slowly-varying dispersion to provide a target for the discrete-unwrapping controller and to estimate water-vapor seeing. The cophasing system is optimized for low control latency, and includes active control of narrowband disturbances. Measurements of the feedback and feedforward transfer functions as implemented are consistent with the predictions of a controls model; on-sky performance data are also presented. 1. INTRODUCTION The Keck Interferometer (KI) 6 (Ragland et al. 2008; Wizinowich et al. 2006; Colavita et al. 2004b) uses several near-infrared fringe trackers (Vasisht et al. 2003) for fringe detection and tracking. The fringe trackers incorporate free-space beam combiners, shared fiber-fed near-infrared cameras, and associated electronics and controls, and operate at H, K, and L bands. The fringe trackers are used to acquire and track fringes using active delay lines; to measure fringe amplitude for visibility science (see Ragland et al. (2008) for a recent overview); and to provide phase referencing for other KI modes including the N band nuller (Serabyn et al. 2004; Colavita et al. 2008, 2009) and the high spectral resolution mode of ASTRA the astrometric and phase referencing upgrade of KI (Woillez et al. 2010). This paper describes the fringe-measurement and control algorithms of the fringe trackers, which have evolved considerably in functionality and performance since first light in 2001 (Vasisht et al. 2003; Booth et al. 2002). Most of the emphasis in the paper is on the recent cophasing implementation for the nuller (Colavita et al. 2008; Booth et al. 2006; Garcia et al. 2006; Colavita et al. 2006), and associated performance improvements. However, we also include some new material on 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA Current affiliation: Sigma Space Corp., Lanham, MD Current affiliation: University of California, Los Angeles, CA Current affiliation: Soliant Energy, Monrovia, CA W. M. Keck Observatory, Kamuela, HI At V 2 calibration and signal-to-noise ratio (S/N) estimation, and quantify some aspects of the underlying fringe-measurement approach. Over the last several years, new near-infrared fringe trackers at other facilities that were designed for cophasing have come on line, in particular the VLTI FINITO three-beam fringe tracker (Le Bouquin et al. 2008) and the VLTI PRIMA fringe sensor unit (Sahlmann et al. 2009). All fringe trackers need to address similar demodulation and control problems: these take slightly different approaches to their solutions, and provide useful perspective, although a detailed comparison of approaches is beyond the current scope. For a broad perspective on optical interferometry, including aspects of instrument design, see the review article by Monnier (2003). Section 2 describes fringe estimation at KI. The high-level approach is similar to that employed at the Palomar Testbed Interferometer (PTI; Colavita et al. 1999; Colavita 1999: henceforth P99 and F99), and uses temporal path length modulation with synchronous array readout to measure the fringe quadratures. The KI implementation uses sliding-window estimation to reduce control latency, which is important for good cophasing performance, and utilizes modified ðs=nþ 2 estimators to provide robust fringe detection in the presence of intensity scintillation. The section also includes bias-correction formulae for V 2 measurements when dewarping is used, extending previous work. Section 3 describes the feedback control implementation. For V 2 measurements, an inner-loop/outer-loop architecture is used, with phase feedback on the inner loop and group delay feedback on the outer loop, coupled with a slow target modulation for reduction of cyclic amplitude systematics. For cophasing, a 795

2 796 COLAVITA ET AL. different servo is used which maintains a fixed phase point within the fringe envelope despite varying dispersion. The section also describes the implementation of the active narrowband control blocks that are used to suppress vibrations beyond the closed-loop bandwidth of the fringe tracker. Section 4 describes the estimation of water vapor turbulence from the fringe tracker observables; these estimates are used to provide corrections to the cophasing commands used for nuller observations, and as an input to the cophasing servo. Section 5 describes the cophasing control implementation, which includes feedforward of fringe tracker residuals to provide improved performance. The section also includes model and measured transfer functions of the feedback and feedforward implementation, as well as sample on-sky data. 2. FRINGE ESTIMATION 2.1. Standard Fringe Measurements The classical ABCD algorithm, as implemented on PTI and other instruments, is used on KI for real-time measurement of phase and group delay, as well as for estimation of fringe visibility for science. With an integrating detector, the flux in the 1=4-wave bins A, B, C, and D is computed from the differences of the detector reads z, a, b, c, and d following an array reset as the fringe is rapidly scanned over one wavelength, viz. A ¼ a z, etc. The fringe quadratures are computed as X ¼ A C and Y ¼ B D, and total flux as N ¼ A þ B þ C þ D. The fringe phase is estimated as ϕ ¼ arctanðy=xþ, and the fringe power as NUM ¼ X 2 þ Y 2, from which the fringe visibility squared is computed, after bias correction of NUM, as V 2 ¼ðπ 2 =2ÞNUM c = N 2, and the photon-noise-limited fringe signal-to-noise ratio squared as ðs=nþ 2 pn ¼ 2 NUM c = N ; see F99 for more detail. Bias correction of NUM is discussed in 2.2, and alternative estimators for NUM and an extended expression for ðs=nþ 2 are described in 2.4. When the length of the path length modulation stroke used to scan the fringe is different from the wavelength, the quantities X, Y, and N, above, differ from their true values. It is convenient, especially for real-time use, to define the dewarping coefficients (see P99) α ¼ 2 sinðs=4þ sinðs=2þ β ¼ 1 cosðs=2þ γ ¼þ2sinðs=2Þ; (1) where s ¼ 2πd=λ, d is the length of the path length modulation stroke, and λ is the wavelength: s ¼ 2π when the stroke length matches the wavelength. Using these coefficients, the corrected quadratures and flux are given by X c ¼ s ðqx þ RY Þ 2π Y c ¼ s ðrx þ QY Þ 2π N c ¼ N þ UðA B C þ DÞ; (2) where Q ¼ 1 β þ 1 α R ¼ 1 β 1 α U ¼ γ=ð2αþ: (3) 2.2. Bias Correction for Power Estimation Squared visibility estimates from the fringe quadratures require correction for biases attributable to photon noise and detector read noise (F99; Perrin & Ridgway 2005). However, after dewarping the phasors to correct for mismatched stroke versus wavelength, there remain cross terms that make the NUM bias phase dependent. We extend the analysis in F99 for this case, below. The expectations on the squares of the corrected quadratures X c and Y c can be written X 2 c s 2 Q ¼ X þ R 2 Y ; (4) 2π R Q or Y 2 c " # X 2 c s 2 ¼ Y 2 c 2π " # Q 2 ð X 2 þ σ 2 A þ σ2 C þ A þ CÞ R 2 " # þ R2 ð Y 2 þ σ 2 B þ σ2 D þ B þ DÞ Q 2 þ 2QRð X Y þσ AB σ BC þ σ CD Þ : (5) The photon-noise variances per bin are shown as A, etc., and the read-noise variances per bin as σ 2 A, etc., each with value equal to the correlated-double-sampling (CDS) read variance σ 2 cds. With an integrating detector, as assumed here, adjacent bins are correlated, viz. σ AB 0, etc., as adjacent bins share a common raw detector read. The only signal-dependent biases in equation (5) are from photon-noise terms. The read-noise biases calibrate simply from the dark values of X 2 c and Y 2 c after correction for detector offsets: call this sum E. Thus the total signal-dependent bias on NUM is 2GN, where G ¼ s 2 1 2π α 2 þ 1 β 2 ; (6) so that an unbiased NUM estimator when using dewarped phasors is given by NUM c ¼ X 2 c þ Y 2 c 2kGN E; (7)

3 FRINGE MEASUREMENT AND CONTROL FOR THE KI 797 where k is the pixel gain in dn per electron. In terms of dewarped quantities only, we substitute N ¼ N c þ πγðy c X c Þ=ð2sÞ, yielding NUM c ¼ X 2 c þ Y 2 c 2kGN c πγ s kgðy c X c Þ E: (8) If s ¼ 2π, then G ¼ 1=2 and γ ¼ 0, and this expression reduces to the standard result (see F99, eqs. [9] [11]) NUM ¼ X 2 þ Y 2 kn 4k 2 σ 2 cds : (9) 2.3. Sliding-Window Estimation For an interferometer observing through atmospheric turbulence with coherence diameter r 0 and wind speed W, the phasedifference coherence time (τ 02 ¼ 0:207r 0 =W) is 4 times smaller than the variance coherence time (T 02 ¼ 0:815r 0 =W) (see P99). In order to maintain adequate continuity between phase samples to allow for unwrapping, we typically require the spacing between samples δt to be τ 02. Similarly, in order to avoid excess fringe blurring during the integration time, we typically require the frame time T to be T 02. For a typical fullframe ABCD implementation, the period of the path length modulation stroke is equal to the frame time, and one phasor estimate is provided per frame. Thus, δt ¼ T, and the frame time is typically limited by the smaller τ 02 constraint on continuity, which impacts instrument sensitivity. The ABCD implementation also affects the performance of the system when the phase estimates are used in a feedforward system to stabilize a second optical path. As discussed in 5, the error rejection for a feedforward system depends strongly on the total control latency Δ, which includes contributions from the fringe measurement, delay-line bandwidth, and data processing. The component of the control latency attributable to the fringe measurement has two components: (a) the data age, i.e., the average delay between when a photon is detected and when a phasor is computed; and (b) the average sampling delay, equal to one-half of the sampling period. Thus the fringemeasurement latency for a full-frame implementation is approximately the frame time: Δ f T. However, for the same frame time, it is possible to reduce the spacing between samples, as well as to reduce the control latency, by updating the phasor estimate as individual detector reads become available, rather than waiting for a full frame s worth of data. For example, when detector read c is available, a new value of the bin count C ¼ c b can be computed. Then using bin counts A and B from the current frame, and D from the previous frame, an updated phasor estimate can be computed using the new information. This approach, slidingwindow estimation, is possible with a sawtooth path length modulation stroke, and is implemented on KI. The KI fringe tracker servo runs at 5 times the frame rate, processing each of the 5 detector reads as they become available. Assume for simplicity an ideal 80% duty cycle path length modulation stroke as illustrated in Figure 1. In this example, at the time of the c read at t ¼ 0:6T, a phasor estimate can be computed from the A, B, C, and D bins centered at 0:1T, 0:3T, 0:5T, and 0:3T, yielding an effective data age of 0:45T. A similar computation for all five reads fz; a; b; c; dg yields data ages of f0:6; 0:55; 0:5; 0:45; 0:4gT. Note that at the time of the z read, no new information is available, and the data age is just 0:2T longer than at the time of the d read. Adding the average 0:1T sampling delay of one half of the sample period, the fringe-measurement latency for the five reads is f0:7; 0:65; 0:6; 0:55; 0:5gT, with an average value Δ f ¼ 0:6T. By comparison, for a full frame readout (at the time of the d read) for this same 80% duty cycle example, the fringe-measurement latency is equal to the data age 0:4T plus the sampling delay of 0:5T, yielding Δ f ¼ 0:9T. In practice, both the sliding-window and full-frame values will be slightly larger when the actual duty cycle and camera readout time are included. We measured the latency at KI from the phase delay between time-tagged fringe-phase measurements and time-tagged delay-line position measurements for a 10 Hz sinusoidal disturbance. The measured value for the sliding-window implementation is 2.6 ms for a T ¼ 4 ms frame time; this is 1:2 ms less than a comparable full-frame implementation would achieve. We complete this section by returning to the issue of phase unwrapping. With the sliding-window implementation, samples are available at 5 times the frame rate, but we need to account for the different data ages between adjacent samples. Continuing the example in Figure 1, the effective delay from the previous sample for reads fz; a; b; c; dg is f0; 0:25; 0:25; 0:25; 0:25gT, with a maximum δt ¼ 0:25T. This is a factor of 4 smaller than for the full-frame approach. This maximum value should also be closely achieved in practice, as additional overheads apply to each sample, affecting mostly the overall latency, and not the sample-to-sample spacing. λ OPD A B C D z a b c d FIG. 1. Example path length modulation stroke and bin timing for slidingwindow fringe measurement. The bin counts are computed from the difference of detector reads as A ¼ a z, etc. The array is reset after the d read during the retrace of the stroke. T t

4 798 COLAVITA ET AL. Thus, with a sliding-window implementation, longer frame times for improved sensitivity can be used while still maintaining adequate phase continuity for unwrapping; or for a fixed frame time, improved unwrapping and feedforward performance can be achieved. For cophasing, KI uses sliding-window estimation with a 4 ms frame time and an update rate of 1250 Hz. For noncophased K-band science measurements, typical frame times are 5 and 10 ms, with update rates of 1000 and 500 Hz, resp Phasor Estimators to Reject Scintillation The fringe power NUM, as computed from the sum of the squares of the fringe quadratures (and in the absence of full-rate photometric monitoring), is biased by scintillation as the frequency of the scintillation approaches that of the path length modulation stroke. At KI, ordinary atmospheric scintillation is usually quite small, because of the near-infrared wavelength and the aperture averaging of the 10 m Keck telescopes. Most of the observed scintillation is caused by fluctuations of the coupling into the fringe tracker single-mode fibers caused by angular vibrations, guiding errors, and Strehl fluctuations. KI implements a multistate fringe-acquisition algorithm, similar to that used at PTI, using filtered and unfiltered versions of the fringe S/N squared ðs=nþ 2 ¼ 2 NUM=ðN þ 4σ cds Þ (10) to transition among states: thus, high levels of scintillation can cause false transitions. Some amelioration is provided by ðs=nþ 2 thresholds that include a fixed term along with a fluxdependent term that sets a lower limit on V 2. However, it is desirable to keep the coefficient of the flux-dependent term as small as possible to allow tracking of extended objects. To reduce the effect of scintillation bias, the real-time system at KI uses an alternative phasor estimator to compute NUM to estimate ðs=nþ 2 using equation (10) for determining mode transitions. We now compare the scintillation properties of the standard phasor estimator and two alternatives Standard Phasor Estimator With the standard phasor estimator, as described in 2.1, the scintillation bias in the fringe power NUM ¼ NUM 0 arises because the quadrature estimates X ¼ A C and Y ¼ B D are essentially derivatives of the incoming flux IðtÞ, i.e., for a stroke period T, XðtÞ IðtÞ Iðt T=2Þ and Y ðtþ Iðt T=4Þ Iðt 3T=4Þ. For this discussion, we assume a full-frame phasor estimator with a 100% duty cycle, and neglect for now the integration over the bin duration T=4. Thus, in the frequency domain, XðωÞ ½1 expð jωt=2þšiðωþ, and for unit scintillation at normalized frequency ω n ¼ 2πf scint T, the bias on NUM 0 is given by S 0 ¼ 2j1 expð jω n =2Þj 2 ¼ 8 sin 2 ðω n =4Þ 1 2 ω2 n; ω n 4: (11) This has an intuitive form, with the bias from scintillation increasing quadratically as the scintillation frequency increases. For comparison with next two estimators, the read-noise bias on NUM 0 is just the last term of equation (9) (assume pixel gain k ¼ 1): E 0 ¼ 4σ 2 cds : (12) For the standard phasor estimator, the standard deviation of NUM in the read-noise limit is just E 0 (see F99). Thus, we use the read-noise bias on NUM as a proxy for the noise on the ðs=nþ 2 estimator Modified Phasor Estimator 1 We can create quadrature estimates that are less sensitive to scintillation by subtracting out a local trend from adjacent bins, i.e., we can compute X 1 ¼ 2½ðB þ DÞ=2 CŠ and Y 1 ¼ 2½B ðaþcþ=2š. With respect to scintillation, these look like centered derivatives, and thus reject low-frequency terms. The bias for unit scintillation on the fringe power estimate NUM 1 ¼ X 2 1 þ Y 2 1 computed from these quadratures is given by S 1 ¼ 2j2 expð jω n =4Þ expðþjω n =4Þj 2 ¼ 32 sin 4 ðω n =8Þ ω4 n; ω n 8: (13) Compared with the standard NUM estimator, the scintillation bias dependence with frequency is now to the fourth power, and the bias drops rapidly with frequency below the corner frequency ω n ¼ 8. However, this improved rejection comes at the expense of read-noise bias. With nondestructive reads, adjacent bins are correlated. Writing NUM 1 in terms of detector reads yields NUM 1 ¼ð a þ 3b 3c þ dþ 2 þðz 3a þ 3b cþ 2. The reset noise drops out in this expression, so that the effective noise variance per detector read is 1=2 of the CDS value, and the bias on NUM 1 is given by E 1 ¼ 20σ 2 cds : (14) Because of its ease of computation it just needs one frame s worth of bin data and its scintillation-rejection properties, this algorithm was used on KI for the first several years in the computation of ðs=nþ 2 for mode transitions.

5 FRINGE MEASUREMENT AND CONTROL FOR THE KI Modified Phasor Estimator 2 An alternative NUM estimator can be created which retains the ω 4 n scintillation rejection of previous version, but with a slightly larger leading coefficient on the bias expression. This estimator uses data from adjacent frames to compute the necessary fringe quadratures. Thus for each new bin count A, B, C, and D, we compute a flux trend by comparison with the prior value of that bin, i.e., on a new A, we compute I 0 ¼ A A last, etc. Thus, in the sliding-window computations, we update the total flux N using the sum of the four most recent bin counts, and update the quadrature estimates for each bin as On a new A, X 2 ¼ A C αða A last Þ=2; On a new B, Y 2 ¼ B D αðb B last Þ=2; On a new C, X 2 ¼ A C αðc C last Þ=2; On a new D, Y 2 ¼ B D αðd D last Þ=2. The factor of 1=2 converts the trend measured over 4 bins to the value for a 2 bin difference, and the factor α is for duty cycle correction; with 100% duty cycle (integration time/frame time), α ¼ 1; more typically, α ¼ 0:8 0:9. The bias, for unit scintillation, for a fringe power estimate NUM 2 ¼ X 2 2 þ Y 2 2 computed using these phasor estimates (assuming α ¼ 1) is S 2 ¼ 1 2 j2 expð jω n=2þ expðþjω n =2Þj 2 ¼ 8 sin 4 ðω n =4Þ 1 32 ω4 n; ω n 4: (15) While not quite as effective at rejecting scintillation as the previous estimator, this NUM estimator does retain the fourth power dependence with frequency, and has much lower noise. On a new D, NUM 2 ¼ðA ðc þ C last Þ=2Þ 2 þðb ðd þ D last Þ=2Þ 2, and thus E 2 ¼ 3σ 2 cds ; (16) which is even smaller than for the standard estimator because of the effective averaging of C and D. The tradeoff for low noise is that the effective frame time is slightly longer than the standard estimator. We find that this NUM estimator works well, and after bias correction using equation (8), is our default in the real-time system for estimating ðs=nþ 2 using equation (10). However, all other phasor computations use the standard estimator. The scintillation bias, for unit scintillation, of the three NUM estimators discussed in this section is plotted in Figure 2. The plot also includes the common factor of sinc 2 ðπf scint T=4Þ to account for integration over the bin duration T=4. 3. FRINGE TRACKING 3.1. Phase and Group Delay Measurement The KI fringe tracker uses two post-combination singlemode fibers on the two outputs of the Michelson beamsplitter to feed an infrared camera, 7 and the camera is typically configured to provide a white-light channel on one output and a spectrometer channel on the other (Vasisht et al. 2003). The white-light channel is typically used for phase estimation for high bandwidth fringe tracking, while the spectrometer channel is used for group delay estimation for central fringe identification. This approach for phase and group delay measurement is similar to that used at PTI. At KI, we use the sliding-window estimator described in 2.3 to provide sets of dark-corrected bin counts A, B, C, and D for the white-light pixel and each spectrometer pixel. With the sliding-window estimator, the computations are done at 5 times the frame rate. For the low spectral resolution modes, we can improve the S/N for phase estimation by combining the white-light and spectral channels to create a synthetic white-light channel. We can create a set of synthetic white-light bin counts by summing the spectrometer bin counts and combining with the white-light bin counts as A 0 WL A WL B 0 WL ¼ B WL 6 4 C 7 WL 5 C 0 WL D 0 WL D WL þ P Aspec P Bspec P Cspec P Dspec : (17) From either the white-light or synthetic white-light bin counts, we compute the complex phasor Γ WL ðnþ ¼X WL ðnþþ jy WL ðnþ, dewarp, and compute the phase as ϕðnþ ¼ arg½γ WL ðnþš, as in 2.1. The raw phase is unwrapped using a first-difference continuity unwrapper with a small amount of memory, viz. ΦðnÞ ¼mod π;π ðϕðnþ Φðn 1Þ N Þþ Φðn 1Þ N ; (18) 7 With post-combination fibers, we do not have separate photometric channels a tradeoff to maximize S/N for phase tracking. For V 2 measurements, our standard observing sequence interleaves 5 sets of single-aperture flux measurements during the fringe scan to approximate simultaneous photometry; the ratio of the average of the single-aperture fluxes provides an estimate of the true flux ratio during the scan, and is used to correct the final V 2 measurement. 8 This form results as we compute the phasors after summing the bins; in terms of phasors, the right-hand side just implements X WL X spec, Y WL Y spec, and N WL þ N spec.

6 800 COLAVITA ET AL NUM Bias f scint T S0 S1 S2 FIG. 2. Scintillation bias vs. normalized frequency for the three NUM estimators. The standard estimator, S 0, has a low frequency dependence of f 2, while the scintillation-invariant estimators S 1 and S 2 have frequency dependencies of f 4, and thus reject low to midfrequency scintillation better. where mod w=2;w=2 ðxþ is the modulo of x over ½ w=2;w=2þ and fðnþ N is an N-point boxcar low-pass filter. Typically N ¼ 2 for this unwrapper. The phase delay xðnþ is just ΦðnÞλ=2π. 9 We compute the group delay gðnþ using the spectrometer phasors Γðn; kþ as a function of wavenumber k. To improve S/N, the spectrometer phasors are phase referenced to the WL phase and averaged as fγðn; kþg ¼ f Γðn; kþγ ðn; WLÞ M g: (19) Typically M ¼ 60, or 50 ms for the nuller configuration. The group delay gðnþ is then computed from the Fourier transform vs. wavenumber of the average, phase-referenced spectrometer phasor array (see P99). The DFT is zero padded by typically 8 times to avoid scalloping losses, and a quadratic interpolation about the peak is done to improve the group delay estimate. KI implements different fringe tracker servos for V 2 and nulling. The purpose of the tracker is not only to center the phase track point near the peak of the coherence envelope, but also to correct for fringe hops in the continuity unwrapper Fringe Tracking with Continuous Centering The servo implementation for V 2 observations uses a fast inner loop and a slow outer loop, as illustrated in Figure 3, to generate servo targets c K for the fast delay line. The inner loop uses the phase delay x with an integral gain a 1 ; a limiter preceding the integrator constrains the maximum slew rate to 9 We tend to use boxcar (FIR) filters for implementing low pass filters at KI, rather than simple fading-memory (IIR) filters; this is a useful robustness measure, as data outliers have a finite lifetime in the filter memory. Our implementation at KI uses simple pushes and pulls from a running average to maintain fixed cost, but with a periodic resynchronization to avoid corruption. prevent fringe blurring. Additional vibration rejection is provided by control block H n, which will be discussed in We typically use an integral gain a 1 ¼ 0:1, which provides, for a 5 ms frame time, a 16 Hz closed-loop bandwidth. It is worth noting that for V 2 measurements, only the motion during a frame time reduces the fringe visibility, and thus high bandwidth is not required: bandwidth is only required to stay within the coherence length of the channel used for the V 2 measurement. The outer loop uses the group delay g with an integral gain a 0 ; typically a 0 a 1 =10, providing an outer loop bandwidth of 1=10 of the inner loop, typically 1.6 Hz for a 5 ms frame time at KI. The outer loop implements the fringe centering, by providing the target for the phase-tracking inner loop. Thus the outer loop corrects for the arbitrary zero point of the unwrapped phase, as well as for fringe hops which may occur in the continuity unwrapper. The correction occurs slowly, at the time constant of the outer loop, so there is insignificant fringe blurring from the changing phase target. By centering continuously, rather than stepwise, no synchronization or coordination of centering commands is needed, and this simplification is one of the advantages of this implementation. The outer loop also accepts an external target. For V 2 measurements, we typically introduce a triangular group delay target modulation, m, with a one-wave peak-to-peak amplitude and a period of 5 s; the period is chosen to be short compared with the length of a science scan. The effect of the target modulation is that the average V 2 for the scan includes measurements over a near-uniform distribution of phases. This greatly reduces the effect of errors that are periodic with a wavelength, such as those 10 We describe only the lock-mode behavior. The system behavior and control of state during fringe acquisition and on loss of lock is important, but beyond the present scope.

7 FRINGE MEASUREMENT AND CONTROL FOR THE KI 801 m a 0 /s H n (s) a 1 /s c K - - g x FIG. 3. Continuous-centering controller; x and g are the estimated phase and group delay, c K is the command to the fringe tracker s fast delay line, and m is a group delay target modulation input. The dotted lines illustrate the logical servo control flow. This controller uses an inner phase loop and an outer group delay loop. The inner loop uses phase feedback and a forward path consisting of an integrator a 1 =s; a limiter preceding the integrator that clips large errors to prevent fast fringe motion which could cause fringe blurring; and a vibrationcontrol block H n ðsþ, which we discuss in 3.4; for now it can be considered a pass-through. The outer loop uses group delay feedback and an integrator a 0 =s to provide a target to the inner loop. The outer-loop bandwidth is typically 1=10 of the inner-loop bandwidth, so that the loops can be analyzed mostly independently, e.g., for analysis of the outer loop, the inner loop can be replaced by its closed-loop response, which is unity at low frequencies. In steady state, the overall controller servos to zero the group delay, and the integral gain in the outer loop corrects for unwrapping errors in the phase delay estimate x (as well as dispersion differences between g and x). For visibility measurements, the target modulation m slowly varies the track point over one wavelength in order to average out phase-dependent systematics. that are attributable to errors in the assumed wavelength, or residual nonlinearities in the path length modulation stroke. It is important to note that while phase delay and group delay are shown as logically equivalent in Figure 3, they differ in practice due to systematic and random dispersion. The systematic dispersion is attributable to average air and material dispersion, with a sidereal component from changing air paths as the delay lines move. The random dispersion is primarily attributable to water vapor seeing (see Colavita et al. 2004a). In the inner-loop/ outer-loop implementation, the dispersion gets pushed into the phase target. As this is a low-bandwidth effect, it is not especially important for V 2. In fact, for V 2, it is the proper behavior, as we wish to track to the center of the coherence envelope, including the effects of dispersion, in order to maximize V 2. g-x u/d/h ± λ B H n (s) a 1 /s c K - - rate limit x g FIG. 4. Discrete-centering controller; x and g are the estimated phase and group delay, c K is the command to the fringe tracker s fast delay line, and g x is a low-bandwidth, nonwrapping estimate of the dispersion. The inner phase loop for this controller is identical to that in Fig. 3. However, the outer loop for this controller generates a target for the inner loop which is constrained to be an integer number of fringe hops. If the dispersion is zero, the outer loop just compares the group delay against a threshold, typically 0:75λ. If the threshold is exceeded, the outer loop steps the target to the inner loop by one wavelength. The low-pass filter B smooths the change in the target to prevent fringe blurring; the time between steps is limited to ensure that previous fringe hops are fully reflected in the group delay before making another hop. Because of dispersion, even in the absence of unwrapping errors, g x, and thus we use as a target for the outer loop a low-bandwidth dispersion estimate g x. This has the effect that the servo follows a fixed phase point on the fringe, even as that point moves within the fringe envelope because of changing dispersion. At the top of the figure is shown the generation of the feedforward command, c N, to the nuller s fast delay line, computed from the sum of the K-band command, c K, filtered by block M, and inner-loop error, e, filtered by block L Fringe Tracking with Discrete Centering For phase referencing, as implemented for the KI nuller, we wish to track to constant phase, rather than to constant group delay, as in the previous V 2 implementation. The implementation we use for phase referencing is shown in Figure 4. In the figure, the inner loop, using the phase delay, remains unchanged from Figure 3. What does change is the source of the phase target, which was previously a continuous target from an outerloop group delay servo, and included dispersion as well as integer fringe hops. In Figure 4, the phase target is now always an integer number of wavelengths, so we always track to zero phase. In principle, we could implement this simply, adding or subtracting wavelengths to the phase target to keep the group delay within 0:5λ. However, with varying dispersion, especially from water vapor seeing, the phase target would continually toggle. For phase referencing, at least for reasonably short tracks where the change in the systematic dispersion due to sidereal motion is less than a wavelength, it is better to follow a single phase point as it wanders within the coherence envelope. For longer tracks, it is necessary to periodically reset the track point in order to not drift too far from the center of the envelope. 11 Thus, in Figure 4, we compare the group delay with a slow, nonwrapping average of the dispersion g x ; we describe how this is computed in 4. A comparator, with thresholds of typically 0:75λ, increments or decrements a counter in units of λ. The output of this counter passes though a short boxcar filter, B, which limits the slew rate of the step, and then to the input of the phase loop; typically B is 20 samples long. The rate at which the wavelength counter can change is limited to allow fringe hops to be fully reflected in the group delay, including the time it takes to fill the group delay coherent average filter, before another hop can occur. Typically, the limit is 100 samples between hops Active Vibration Cancellation With typical closed-loop bandwidths of 16 Hz, the basic fringe tracker has limited ability to attenuate high frequency disturbances. However, at KI there are residual disturbances at high frequencies that are not observed by the laser metrology or accelerometers. Most of these disturbances are narrowband vibrations, and typical sources are induction motors which introduce 11 We readily see this effect for long tracks, manifesting itself as a smooth decrease in white-light V 2 with time. e L M c N

8 802 COLAVITA ET AL. H(ω) ω 1 cos(ω 0 t+φ 0 ) ω 0 X X sin(ω 0 t+φ 0 ) 2k 0 s+δ 2k 0 s+δ cos(ω 0 t) X X sin(ω 0 t) FIG. 5. Active vibration-control block, showing one of several HHC controllers, and the pass-through path on the bottom. The HHC controller in the figure is tuned to frequency ω 0 : it mixes the frequency band near ω 0 to baseband, applies an integral controller, and then mixes it back up to the original frequency. This provides gain over a narrow frequency range which is used in the inner servo loops of Figs. 3 and 4 to suppress narrowband disturbances. vibrations at 29 and 58 Hz (for US 60 Hz ac power), as well as mystery sources at other frequencies. For such narrowband disturbances, it is possible to actively suppress them to improve fringe tracker performance, even though their frequencies exceed the ordinary fringe servo closed-loop bandwidth. The control block used for vibration suppression is shown in Figure 5, and implements higher harmonic control (HHC; Shaw & Albion 1981). This block is inserted in the fringe tracker forward path as shown in Figures 3 and 4, and provides loop gain at the frequency of the vibrations. As shown in Figure 5, the overall control block has a pass-through path on the bottom, and multiple HHC blocks for each vibration frequency under control. While each control block typically controls five frequencies at KI, we will generally assume a single controlled frequency for the rest of this discussion. In Figure 5, quadrature oscillators configured to the nominal vibration frequency mix the input signal to baseband. The phase shift ϕ 0 on the input oscillators compensates for phase shifts in the forward path of the fringe tracker at the frequency of the vibration, including the phase of the ordinary fringe tracker integral controller. The controllers on the baseband disturbance signals are implemented as leaky integrators with gain k 0. The small leak δ is so that the oscillators decay to zero when the fringe tracker enters a hold state on loss of lock, although it does limit the ultimate vibration rejection. The integrator outputs are then mixed back up to the original frequency and summed with the pass-through signal. The response of the HHC controller is given by Hall & Wereley (1989; summarized in Sievers & Von Flotow 1992). Including the pass-through, and the finite leak in the HHC integrator, the transfer function of the vibration control block is H n ðωþ ¼1 þ 2k 0ðaðs þ δþþbω 0 Þ ðs þ δþ 2 þ ω 2 ; (20) 0 where a ¼ cos ϕ and b ¼ sin ϕ. Thus, H n ðωþ provides high gain at the configured frequency, and is approximately unity elsewhere. At KI, H n ðωþ is used in series with an ordinary servo controller GðωÞ. While the approach is generalizable, we are normally interested in providing suppression above the unity-gain frequency of the ordinary servo, i.e., where G ¼jGðω 0 Þj 1; we assume G incorporates any dynamics of the plant. In this case, near the configured frequency, the second term of equation (20) is 1, and so we can approximate H n ðωþ near ω 0 as k H n ðωþ 0 e jϕ jðω ω 0 Þþδ : (21) In our application, we typically set ϕ arg½ðgðω 0 ÞŠ. The 3 db half width of these active notches Δω, for Δω δ, depends on the product of the HHC gain and the ordinary loop gain as Δω Gk 0 rad=s, with peak suppression Δω=δ. To first order, the maximum notch width is limited by the requirement that the ordinary controller and plant be approximately constant in phase and amplitude over the notch width in order to control the phase at the loop crossovers on either side of the notch. This constraint is similar to that which limits the bandwidth of the ordinary controller, and thus the notch widths are typically limited to a fraction of the ordinary closed-loop bandwidth. We find these modules to be robust and effective in suppressing vibrations with well-known frequencies, which is the case for the largest-amplitude vibrations at KI. 12 The HHC approach also allows for accurate single-parameter tuning of the notch frequency. For a different approach to narrowband suppression, see Di Lieto et al. (2008), which describes the adaptive algorithm used on VLTI s FINITO fringe tracker. As the ordinary controller gain, while less than unity at the frequencies where these notches are employed, is still finite, and there are multiple notches which can interact, we usually tune the notches empirically. In our implementation, we include a test oscillator in the HHC block to measure arg½ðgðω 0 ÞŠ, and we set the notch gain based on frequency-response measurements in order to balance notch depth with band-edge peaking. Typically bandwidths of order 1 Hz are used, which allows for some uncertainty in the vibration frequency, e.g., to account for shifts in the frequency of induction motors with changing loads. 12 We have also used them in other subsystems of KI (see Colavita et al. 2004b); the current version is adapted from one originally coded by E. Hovland for tilt control.

9 FRINGE MEASUREMENT AND CONTROL FOR THE KI 803 For example, with a 1 Hz notch width, and δ 1 Hz, a factor of 4 suppression is possible for uncertainties in the vibration frequency of 0:125 Hz. 4. DISPERSION ESTIMATION One of the modes of KI is broadband nulling at 10 μm (see Colavita et al and references therein). Because of the integration times required to obtain adequate S/N, high bandwidth path length control to correct for rapid atmospheric turbulence is not possible using just the 10 μm light, and thus the high bandwidth control is provided by the fringe trackers using the K-band light. Because of atmospheric dispersion, simple achromatic feedforward is not adequate when high accuracies are required, and a correction for the effects of atmospheric water vapor is also required (Colavita et al. 2004a; Colavita 2010). The path length fluctuations attributable to water vapor can be estimated from the relative group delay, and can be used with a dispersion model to correct the N-band path length residual attributable to water vapor. In principle, the relative group delay is just the difference of the group delay g and the phase delay x described in 2.1. In practice, the phase delay x is subject to unwrapping errors, caused by, e.g., fast fringe motion or signal dropouts, and a more robust estimator is desired, especially for low S/N conditions. The algorithm used at KI takes advantage of the fact that the water vapor coherence time is much longer than the dry air coherence time. We start by averaging the phase delay x with a filter that has the same length as the group delay coherent averaging filter in equation (19): call this value x 0, so that x 0 has the same latency and smoothing as g, and compute Λ ¼ g x 0. If there were no unwrapping errors, Λ would have a Gaussian-like distribution about its true value, and we could simply average Λ to improve its S/N. However, if the true value of Λ is changing slowly over the desired averaging times, we can average the central modulo in order to be robust against unwrapping errors in x. Because of the width of the distribution, which can exceed one wavelength under low S/N conditions, we need to choose the offset about which to do the central modulo to center the distribution in the interval. In this case, the average of the wrapped quantity will suffer the least bias from the tails of the distribution. As an extreme example of the type of bias that can arise, assume that the true value of Λ is slightly less than λ=2. If we do a central modulo about zero, the wrapped distribution will be peaked near λ=2, and a simple average will yield a value near zero, instead of the true value. A simple algorithm for selection of the offset about which to do the central modulo is to choose the one that minimizes the mean square of the distribution, and this is what we implement. There are certainly other approaches that could be used, but this is computationally efficient in our system. Thus, for each time sample, for a set of offsets fp i g, typically f 0:5; 0:375; 0:25; ; 0:375gλ, we compute the differences fδ i ¼ mod λ=2;λ=2 ðg x 0 p i Þg, averages fd i ¼ δ i þ p i g, and mean squares fs i ¼ δ 2 i g. Typical averaging times are 1.0 s. If c is the index of the offset that gives the minimum value of s i, our estimate of the average dispersion is just d c. In practice, we do some (probably unnecessary) interpolation, and compute a three-point quadratic fit to the mean square about offset p c to determine an optimal offset, and then interpolate between the nearest d i to determine d c.asd c is a wrapped quantity, we run it through a first-difference continuity unwrapper with a small amount of memory (typically 1 s) to yield our final dispersion estimate Λ. b This value is used to compute the nuller water-vapor feedforward quantities (Colavita 2010), and to provide the zero point for the outer loop for the cophasing (discrete-centering) fringe tracker in Figure 4. Strictly, the zero point of Λ b is somewhat arbitrary. However, changes from this zero point are correctly measured, and this is all that the nuller requires, as the nuller is subject to a number of different zero point offsets, which it determines at low bandwidth directly from the 10 μm fringe signal. In particular, for control of the nuller s longitudinal atmospheric dispersion corrector, Λ b is high pass filtered at Hz, and we rely upon direct 10 μm feedback to control the very low frequencies. 5. FEEDFORWARD CONTROL As discussed previously, one role of the K-band fringe trackers is to provide (coaxial) phase referencing, or cophasing, for the nuller. To first order, as the nuller and fringe tracker have separate delay lines, one could just slave the nuller delay lines to the K band delay lines. However, as the nuller delay lines are outside of the fringe tracker feedback loop, the K-band fringe tracker error can be used in addition to the K-band feedback signal to generate commands to the nuller delay lines to improve the cophasing performance without violating stability criteria (Lane & Colavita 2003). The implementation at KI is shown schematically at the top of Figure 4, where c K is the command to the K-band delay lines from the discrete controller and e is the K-band fringe servo error. These two inputs are filtered by blocks M and L, and summed to generate the command to the nuller delay lines, c N. To first order, blocks M and L are just pass-throughs, e.g., M ¼ L ¼ 1; in practice, they can be used optimize performance, and in our implementation L is a lead filter, and M is a short boxcar filter. A simple controls model of the feedback and feedforward systems is given in Figure 6, where d is the disturbance input, e FB is the feedback error (seen by K band), and e FF is the feedforward error (seen by N band). In the figure, block D models: the overall system time delay, Δ, including the fringe tracker measurement latency Δ f as discussed in 2.3; the delay line command response; and compute and transport delays. Block D also includes the amplitude effect of the finite fringe tracker frame time T, viz.

10 804 COLAVITA ET AL. d DðfÞ ¼sincðπfTÞ expð j2πfδþ: (22) Block G models the feedback controller; typically G is just an integrator: GðfÞ ¼a=ðj2πfÞ. With these definitions, the feedback error response takes the usual form e FB d ¼ 1 1 þ DG ; (23) which at low frequencies with integral control has the asymptotic form e FB d g=f: (24) Blocks M and L in Figure 6 are the same as in Figure 4, where M filters the K-band delay line command, and L filters the fringe tracker error. The feedforward error response for this system can be written e FF d 1 D½L þ GðM 1ÞŠ ¼ : (25) 1 þ DG In this expression, if the gain G ¼ 0, or else M is chosen to compensate for the time delay, viz. M ¼ LD, then e FF d ¼ 1 DL; (26) independent of the feedback gain. In this case, if L ¼ 1, and neglecting the amplitude effect of the sinc function in equation (22), this becomes a time-delay limited response, and the response as a function of frequency f is given by e FF d ¼ 2 sin πfδ: (27) If instead M ¼ 1, then - e FB D FIG. 6. Servo performance model. Blocks D and G model the forward path of the feedback controller, and e FB is the feedback error in response to disturbance d. The feedforward command is computed as in Figure 4 from the feedback-loop output and error filtered by blocks M and L. The feedforward error, e FF, is the difference of disturbance and the feedforward command. This model is used for the performance analyses in the text. L G - M e FF e FF d ¼ 1 DL 1 þ DG ; (28) and the feedforward error response is the product of the timedelay limited response (eq. [26]) and the feedback response (eq. [23]). In this case, the low frequency-error response falls off as frequency squared for an integral feedback controller. This behavior is desirable to provide good cophasing performance in the presence of the steep atmospheric power spectrum. As noted previously, blocks L and M can be used to improve performance. For block L, we usually implement a lead filter, which can compensate some of the system latency at low frequencies, in exchange for some peaking at higher frequencies (but where the response is attenuated by the low pass filtering of the finite frame integration time included in equation [22]). For block M, we usually introduce a small amount of delay, Δ M, implemented as a short boxcar filter (the amplitude response is not significant over the frequency range of interest). This delay reduces some midfrequency peaking attributable to the interaction of the numerator and denominator in equation (28), but limits the ultimate low frequency rejection. This can be seen by letting G in equation (25): the low frequency error response becomes time-delay limited as 1-M, with error response given by equation (27) with Δ ¼ Δ M. Figures 7 and 8 show model and measured performance for the cophasing system at KI. The measurements were made by introducing synchronized disturbance targets on the K- and N- band delay lines, and measuring the residuals when the fringe tracker was tracking an internal source. All data is for a frame rate of T ¼ 4 ms, and a sample rate of 1250 Hz for our 5 point sliding-window implementation. Control blocks G, L, and M were implemented as: Feedback controller G: Integral controller with a 15.9 Hz crossover (dimensionless gain 0.08); Error filter L: Lead filter with a zero at 133 Hz (and an implicit pole at the 625 Hz Nyquist frequency; at low frequencies, this filter looks like a 1 ms time lead); Feedback filter M: Time delay with Δ M ¼ 0:8 ms (implemented as a nonbuffered 3-point boxcar filter). For the system delay in block D in the model, we used Δ ¼ 4:0 ms, consisting of 2.6 ms for the fringe measurement, discussed in 2.3, 0.7 ms for the measured delay line command response, and 0.7 ms for compute and transport delays. 13 Figure 7 shows the measured feedback and feedforward responses versus the models: the compliance is fairly good. The feedforward performance is better than the feedback performance below about 60 Hz, with the error decreasing rapidly with decreasing frequency until the limit imposed by the delay Δ M comes in. At this point, the error rejection continues to 13 As part of the overall preparations for the nuller mode, the delay line servo was optimized to improve the linearity of the phase of the command response for use in this feedforward application. In addition, a special low-latency control input was added to accept the output of the lead filter L.

11 FRINGE MEASUREMENT AND CONTROL FOR THE KI Feedback data Feedforward data Feedforward model delay = Feedback model gain frequency (Hz) FIG. 7. Model feedback and feedforward error response and measured data; a 4 ms time-delay-limited response is included for reference. The error response with feedforward is better than with feedback below 60 Hz. improve, but only linearly with decreasing frequency. In the critical Hz region, the feedforward response can be modeled as a simple 4 ms time delay. At lower frequencies, the performance is substantially better than a simple time delay due to the contribution of the feedback component. The parameters used here of course represent a compromise: with a lower value of the integral gain, or a larger compensating delay Δ M, the feedforward response would approach a delay-limited system with a time delay of approximately 3 ms. The larger effective midfrequency time delay in our implementation is in exchange for improved low frequency rejection and maintenance of good feedback performance. Figure 8 is like Figure 7, except now with three active notches enabled, as described in 3.4. The notches add a small Feedback data Feedforward data Feedforward model delay = Feedback model gain frequency (Hz) FIG. 8. Model feedback and feedforward error response and measured data with 3 notches enabled; a 4.5 ms time-delay-limited response is included for reference. The broadband performance is similar to Fig. 7, but with slightly more high-frequency latency. The notches provides addition rejection at selected frequencies.